Yeah, at level 15 you're just hitting the wall where two-hander attack speed becomes a real liability and you don't have the perks yet to fix it. Dual-wield feels snappier because it literally IS snappier - you're getting two animations per swing, so even if each individual hit is weaker, the rhythm is way tighter and you can react faster to incoming damage.
The good news: greatsword doesn't suck, you're just building it wrong for early game. Grab the perk that increases two-hander attack speed (should be in the second or third tier of the two-handed tree) and suddenly it feels way more responsive. There's also a later perk that staggers on power attacks, which turns greatsword into a crowd control tool instead of just a slow damage dealer. That changes how you actually play - you're not trying to duel draugr, you're interrupting them.
That said, dual-wield IS more forgiving on Legendary early on because you get more chances to dodge between swings. If you're getting punished constantly, that's valid. You could also just... switch weapons based on enemy type? I do that on my Legendary runs - greatsword for single tough enemies, dual-wield for groups. Not optimal for speedrunning but way less frustrating.
What difficulty are you actually on? If it's Legendary and you're genuinely stuck, sometimes the answer is just "come back to greatsword at level 25 when you have the perks" rather than forcing it now.